The zero-one law for a complete orthonormal system

Kazaros Kazarian

doi:10.1016/j.crma.2004.07.009

Mathematical Analysis

The zero-one law for a complete orthonormal system

Presented by: Jean-Pierre Kahane

Kazaros Kazarian ¹

¹ Departamento de Matemáticas,C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 335-337.

Abstract
Résumé

A complete orthonormal system of functions $Θ = {θ_{n}}_{n = 1}^{\infty}, θ_{n} \in L_{[0, 1]}^{\infty}$ is constructed such that $\sum_{n = 1}^{\infty} a_{n} θ_{n}$ converges almost everywhere on $[0, 1]$ if ${a_{n}}_{n = 1}^{\infty} \in l^{2}$ and $\sum_{n = 1}^{\infty} a_{n} θ_{n}$ diverges a.e. for any ${a_{n}}_{n = 1}^{\infty} \notin l^{2}$ . We also show that for any complete ONS ${f_{n}}_{n = 1}^{\infty}$ of functions defined on $[0, 1]$ there exists a fixed non decreasing subsequence ${n_{k}}_{k = 1}^{\infty}$ of natural numbers such that for any $f \in L_{[0, 1]}^{0}$ and some sequence of coefficients ${b_{n}}_{n = 1}^{\infty}$ ,

\sum_{n = 1}^{n_{k}} b_{n} f_{n} \to f a.e. when k \to \infty .

On construit un système orthonormal complet $Θ = {θ_{n}}_{n = 1}^{\infty}, θ_{n} \in L_{[0, 1]}^{\infty}$ tel que $\sum_{n = 1}^{\infty} a_{n} θ_{n}$ converge presque partout pour n'importe quel ${a_{n}}_{n = 1}^{\infty} \in l^{2}$ et diverge presque partout pour n'importe quel ${a_{n}}_{n = 1}^{\infty} \notin l^{2}$ . Nous démontrons que pour toute système orthonormal complet ${f_{n}}_{n = 1}^{\infty}$ il existe une sous suite croissante ${n_{k}}_{k = 1}^{\infty}$ d'entiers naturels tels que pour tout $f \in L_{[0, 1]}^{0}$ il existe une suite de coefficients tels que

\sum_{n = 1}^{N_{k}} b_{n} f_{n} \to f p.p. si k \to \infty .

Received: 2004-06-17
Accepted: 2004-07-12
Published online: 2004-08-14

DOI: 10.1016/j.crma.2004.07.009

Author's affiliations:

Kazaros Kazarian ¹

¹ Departamento de Matemáticas,C-XV, Universidad Autónoma de Madrid, 28049 Madrid, Spain

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     author = {Kazaros Kazarian},
     title = {The zero-one law for a complete orthonormal system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {335--337},
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     number = {5},
     year = {2004},
     doi = {10.1016/j.crma.2004.07.009},
     language = {en},
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Kazaros Kazarian. The zero-one law for a complete orthonormal system. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 335-337. doi : 10.1016/j.crma.2004.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.009/

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